Final answer:
To determine the most profitable quantity and maximum profit, subtract the total cost from total revenue to obtain the profit function, find its critical points by setting its derivative to zero, and calculate the profit at these points.
Step-by-step explanation:
Calculating Maximum Profit and Output Quantity
To calculate the quantity of output that will provide the highest level of profit, we need to determine the difference between total revenue and total cost. This is done by subtracting the cost function from the revenue function. For the given revenue function R(x) = 132x - 0.0135x2 and cost function C(x) = 8000 + 66x - 0.027x2 + 0.00001x3, we calculate the profit function P(x) = R(x) - C(x).
To find the number of units that will give maximum profit, we would set the derivative of the profit function P'(x) equal to zero and solve for x. This gives us the quantity at which profit is maximized. The critical point found must be a maximum by verifying it is a peak on the profit curve (e.g., using the second derivative test).
To find the maximum possible profit, we substitute the x value found from the first step into the profit function P(x). The resulting value indicates the highest achievable profit for the given product, assuming that all units produced are sold.